@John – thanks for that, I remember proving that Eckmann-Hilton result about two binary operations on the same set in an exercise many years ago, but I had no idea about its significance or that it had a name!

I was thinking that the "swap round the middle two terms" line in that proof reminded me of how you prove that \\(f + g\\) is a homomorphism given homomorphisms \\(f\\) and \\(g\\) of Abelian groups (or commutative monoids more generally):

\\[(f + g)(x + y) = f(x + y) + g(x + y) = fx + fy + gx +gy = fx +gx +fy + gy = (f + g)x + (f + g)y\\]

It strikes me that the Eckmann-Hilton argument might be some kind of converse to this, but I can't quite see how.

I was thinking that the "swap round the middle two terms" line in that proof reminded me of how you prove that \\(f + g\\) is a homomorphism given homomorphisms \\(f\\) and \\(g\\) of Abelian groups (or commutative monoids more generally):

\\[(f + g)(x + y) = f(x + y) + g(x + y) = fx + fy + gx +gy = fx +gx +fy + gy = (f + g)x + (f + g)y\\]

It strikes me that the Eckmann-Hilton argument might be some kind of converse to this, but I can't quite see how.